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The transverse momentum $j_{\rm T}$-dependent jet fragmentation functionshave been investigated in proton+proton (p $+$ p) and proton+lead (p $+$ Pb)collisions at $\sqrt{s_{\rm NN}} = 5.02~{\rm TeV}$ with a multiphase transportmodel containing both the simple quark coalescence mechanism and a new hybridhadronization mechanism with coalescence and fragmentation processes.Hadronized by the new hadronization mechanism, the AMPT model achieves aquantitative description of the $j_{\rm T}$-dependent jet fragmentationfunctions measured by ALICE. Besides, no obvious jet-medium interaction andcold nuclear matter effects on the $j_{\rm T}$-dependent jet fragmentationfunctions in p $+$ Pb collisions were observed. We found the $j_{\rmT}$-dependent jet fragmentation functions are dominated by the quarkcoalescence contribution for the new hadronization mechanism, which can bedecomposed into narrow and wide parts. The root mean square value of the widepart depends on the jet radius $R$ and jet transverse momentum $p_{\rm{T,jet}}$, which is sensitive to different hadronization mechanisms and theircomponents. Therefore, the $j_{\rm T}$-dependent jet fragmentation functionsare proposed as a sensitive probe to study the non-perturbative hadronizationeffect of jets in small colliding systems.

Hadronization effects on transverse momentum dependent jet fragmentation function in small systems